Cooperation with: P. Dai Pra (Universitŗ degli Studi di Padova, Italy), D. Dereudre (…cole Polytechnique, Palaiseau, France), M. Sortais (Berliner Graduiertenkolleg ,,Stochastische Prozesse und Probabilistische Analysis`` (Graduate College ``Stochastic Processes and Probabilistic Analysis'')), M. Thieullen (Universitť Paris VI ``Pierre et Marie Curie'', France), L. Zambotti (Scuola Normale Superiore di Pisa, Italy, and Technische Universitšt Berlin), H. Zessin (Universitšt Bielefeld)
Supported by: Alexander von Humboldt-Stiftung (Alexander von Humboldt Foundation): Fellowship
In the following works, we are interested in the analysis of several types of interactive diffusions modeling phenomena coming either from Statistical Physics or from Population Dynamics. The underlying idea is to transpose important concepts and tools from Statistical Mechanics like Gibbs equilibrium measures, entropy, space-time limit, to mathematical objects like diffusions, or Brownian semi-martingales. Working on path space, we also obtain a better understanding of the behavior of such diffusions.
Together with M. Thieullen we study the class of all probabilities on the path space which have the same bridges as the following -valued reference Brownian diffusion denoted by Pb and law of the solution of the stochastic differential system:
where the drift is a regular function on .
Equation (1) is a perturbation of the duality equation
satisfied by Brownian bridges, duality between the
Malliavin derivation operator and the stochastic integral. The
perturbation terms (second and third terms in the RHS of equation
(1) ) are to be compared with the Malliavin derivatives of
Hamiltonian function associated to Gibbs measures. The main difference from
the one-dimensional situation studied before in 
comes from the new last term in (1), the stochastic integral
the reciprocal characteristic G w.r.t. the
coordinate process. This term vanishes if and only if the drift b of the
Brownian diffusion is a gradient. In  this term was
identically zero since each regular function is a gradient in dimension
One finds in  several applications of this characterization of reciprocal processes. Among others, the authors prove a generalization of the famous Kolmogorov theorem about reversible diffusion: the existence of a reversible law in the reciprocal class of a Brownian diffusion with drift b can only occur if b is a gradient. In collaboration with L. Zambotti, they also study an application of the duality equation (1) in the singular case of -Bessel processes. There the difficulty consists in generalizing the reciprocal characteristics F and G associated to the degenerate drift function .
The infinite-dimensional situation, when the index i gets continuous () and the reference process is solution of a stochastic partial differential equation, is the next step in the study of such topics. It will be the subject of a forthcoming paper.
With D. Dereudre  we study
Gibbsian properties on the path level of continuous systems of
interactive Brownian diffusion. More precisely we consider the law of the solution of
the following stochastic differential system with values in :
In collaboration with P. Dai Pra and H. Zessin , we give a
Gibbsian characterization for the stationary law of the interacting
defined on the lattice and solution of the following type
stochastic differential system
In , based on the Gibbsian characterization shown in , a weak existence result for the solution of (3) is proved. The authors use a space-time cluster-expansion method, which is powerful when the coupling parameter is sufficiently small. As conclusion, the Gibbsian approach to study infinite-dimensional processes seems to be very effective in situations where the stochastic calculus can not give an answer, like for example in the question of existence of solution for the non-Markovian equation (3).
Consider now the following model of Statistical Physics in random
Langevin dynamics for a ferromagnetic
system submitted to a disordered external Bernoulli magnetic field:
In  the authors use the cluster-expansion method to show that the space-time correlation functions associated to Y decay exponentially fast in the high-temperature regime. In particular, they prove that for small enough, the infinite-dimensional process Y(t) is ergodic and the velocity of the convergence is exponential. The specific difficulty of this model (compared to the system (3)) comes from the fact that the non-Markovian interaction (last term in the RHS of (5)) is no more local in time: it depends on the values of Y on the full time interval [0,t].
The process studied in  arises as diffusion
limit of branching particle systems. It is called
catalytic super-Brownian motion, in the sense that the branching procedure
in the approximation depends strongly on a singular medium called
catalyst. It is modeled as solution of the following stochastic partial